Chapter 5 Lecture 10

Spring 2015

Nonlinear Elements

• 1. A nonlinear resistance • 2. A nonlinear reactance• 3. A time varying element in you circuit or

system. • 4. These elements show up in many form and

the biological ones are more complicated than the electronic ones.

Basic Characteristic of NonlinearDevices.

• 1. Nonlinear resistance,

An Ideal Harmonic Generator

• 1 The simple one is a diode. I= Vo+αV1+βV2+----

• V

Test Circuit

Results

Nonlinear Reactance

• 1. Use to convert power from one frequency to another.

• 2 Typical diode C~(V)-1/2 for step diode• 3 How do you design a diode with a larger

nonlinear capacitance?• P-_ N_ P+ N+ Ni N+

Parametric Amplifiers

• 1. Conservation of Energy on a photon basis

• 2. Conservation of momentum where k is the propagation constants

Parametric Amplifiers

Biological Amplifiers

• http://www.unmc.edu/physiology/Mann/mann13.html

• Neural Transmitter Releases up to 104 calcium ions• Need to overcome the electrical threshold for firing

Stochastic Resonance

Stochastic Resonance

12

Nonlinear Effects at Cell Membranes

1. Current flow for

2. Rm is the membrane resistance. The result is that the membrane is a poor rectifier. However AC voltages make the interior more negative.

AC Induced Current Flows At Low Frequencies

Induced DC Currents for VAC from -60 to + 40mV

For a spherical cell.

Shift in Membrane Firing Time

• Shift in firing time for• Where u(t) is unit step function

Mode Locking of Oscillators

• Theory for injection locking of electronic oscillators is give by

• The theory is good for case where

• This worked for Aplysia pacemaker cells.

Threshold Injection Lockingfor an Aplysia Pacemaker Cell

• Frequency range from 2 to 10 Hz

Signal Noise Requirements for Phase Locking

• The phase of the inject signal must be stable enough so that the phase φ

• Where K is the linear control characteristic in units (2π Hz/V) and is closely related to the loop gain.

Locking of a Pacemaker Cell

• Response to various frequencies of injected currents.

Signal Coherence

• Litovitz showed that for 10µT coherence for 10 seconds or longer was required for signals at 55 or 65 Hz was required to change the activity of

• τcell= 8 sec

sec8cell

Litovitz shows both space and time coherence help separate signals from Noise

Results Show

• 1. Both Space and time Coherence are important.

• 2 Small electric fields can lead to biological changes.

• 3. Magnetic fields can affect biological changes by a separate mechanism.

Effects of Time Delay Between E and J

• This can give Z in all four quadrants.

Membrane Capacity as a Function of Frequency

• Membrane Capacity is only a small function of voltage.

Repetitive Stimulation

• 1. Repetitive microwave pulse resulted in decreasing the amount of slowing for a pacemaker cell in Aplysia.

• 2. Repetitive electrical stimulation lead to decreases in the resistance of gap junctions and to a 62% increase in coupling between cells.

• 3. These are likely to be the result of feedback leading to adaptive responses.

A Neural Network Model for Adaptive Responses

• 1

Training to Recognize 60Hz as a Function of S/N with 97% Accuracy

Thermal Calculations

• Power in and rate of change of temperature

Maximum Temperature change for a small sphere with total energy in H

Thermal Chemistry • S = fraction that under gone chemical change• K’ is the chemical reaction rate. • R’ is the gas constant• H’ is free energy , S’ is the entropy.

• This leads to an exponential of an exponential

Thermal Chemistry

• 1. Rule of thumb we are likely to see biological changes when

• 2. The body typically holds your temperature to +/- 0.5oC

• 3. Very rapid changes in chemical reaction rates above a threshold. !!

The Rate of Change of Temperature is also Important.

• 1 We have shown the changes of 1/10 oC can change the firing rate of a pacemaker cell at 1oC/sec.

• From the Nernst Equation

• Slow increases in T increased firing rates of a pacemaker cell rapid one decreased it. Changes seen with as little as 0.1 o C at rates of 1o C/sec

Effects of Rapid Heating

• Picture from • Aplysia

Discussion

• 1. It takes high powers and short pulses to get significant temperature differences on small objects.

• 2. Thin films have larger surface to volume ratios and cool faster than spheres .

• 3. Blood flow cools hot spots. • 4. The thermal time constant is an important

parameter and the sensitive to temperature change is one of the first measurements to make on any experiments involving RF or Microwaves.

Discussion

• 1. Temperature pulses lead to thermal expansion and can cause acoustic waves that can be sensed at a distance.

• 2. Example radar hearing.

Natural and Man-Made Fields

• 1. The atmosphere charged about 100/sec world wide with about an 18 sec time constant to about 130V/m

• 2. Peak values at about 3000V/m• 3. Rapid decrease with frequency to typical

value > 1 Hz of 10-4 V/m• 4. These numbers are all variable

Internal Fields

• 1. Across a membrane of 2 x 107V/m• 2. Nerve pulses about 0.4ms , rise time 0.1ms

fall time 0.5ms. Dead space 1 to 3ms• 3. Fields along the outside of a nerve cell • 5x10-2V/m• 4. These numbers are variable with position,

type of cell etc.

Types of Noise

• 1. Thermal • 2. Shot Noise • 3. C/fn Noise• 4. Noise generated by other electrical activity

in the Body.

Thermal Noise.

• 1. Pn= kTB = kTΔf• 2. Other forms for matched loads

• 3 For thermal equilibrium. Non-equilibrium get negative temperatures.

Spontaneous Emission and Shot Noise

• 1 Spontaneous Emission • P= hfΔf • 2. Shot Noise

• 3. 1/f Noise or

• Where S(f) is the power spectral density

Example

• 1. For mylar film

Membrane Example.• 1

Other Electrical Activity

• 1. EEG• 2. ECG or EKG• 3 Muscle movement.• 4. Nerve Cells Firing

Minimum Detectable Electric FieldIs a Function of Frequency

• Bovine Fibroblast• Cells

I= 10-3—10A/m2